Child-resistant packages are conventionally employed for prescription vials, vitamin bottles, and a number of other applications including containers for caustic or hazardous materials. The present invention pertains particularly to those types of child-resistant packages that involve application of axial pressure and simultaneous turning of the closure with respect to the container in order to remove the closure from the container. The present invention involves a number of features or aspects in a child-resistant closure, container or package, which may be implemented separately from, or more preferably, in combination with each other.